Mohammad Hadi Hedayatzadeh (Institute for Research in Fundamental Sciences, Tehran)

In this talk, I will present a joint work with O. Bültel and my student A. Partofard on prismatic windows with additional structures. I will start with a brief overview of the theory of displays, developed by Th. Zink, which is a generalization of Dieudonné theory. Displays play a crucial role in the study of Barsotti-Tate groups when the base is not a perfect field of positive characteristic. Zink has further developed the theory and introduced windows over frames. In another direction, in order to construct integral models of Shimura varieties that are not of Abelian type, O. Bültel defined and studied displays with additional structures, called (G,μ)-displays. In this joint work, we combine these two inventions to define and study (G,μ)-windows and show that under some mild conditions, the category of (G,μ)-windows is equivalent to that of (G,μ)-displays. Finally, in a joint project with Partofard, we develop the theory of prismatic windows, which is better adapted to the setting of pefectoid geometry and is closely related to the stack of G-torsors over the Fargues-Fontaine curve.